You will need to replace the crank to go from a regular (53,39) double to a compact (50,34) double. The bolt circle of standard cranks is too large to fit the smaller inner chainring of a compact crank. You should not need to change the chain, since it should only need to be shortened. You will need to move the front derailer to line up with the smaller chainrings of the compact crank.

Going from a standard double to a compact double requires a new crankset (110 BCD= compact) and matching bottom bracket. Front derailleur will need to be adjusted and in some cases the rear derailleur will need to be replaced depending on the chainwrap capacity needed for the particular crankset and cassette if you plan to run a wide ratios.
I would consider a 50-36 compact instead of the more common 50-34 to avoid wide ratios in the front, that is if the 34 would give you low enough gearing. Or a 48-34 compact would give you overall lower gearing and closer ratios but there could be problems getting the front derailleur low enough if it is a braze-on type.